On the use and error of approximation in the Domenico (1987) solution

A mathematical solution for solute transport in a three-dimensional porous medium with a patch source under steady-state, uniform ground water flow conditions was developed by Domenico (1987). The solution derivation strategy used an approximate approach to solve the boundary value problem, resulting in a nonexact solution. Variations of the Domenico (1987) solution are incorporated into the software programs BIOSCREEN and BIOCHLOR, which are frequently used to evaluate subsurface contaminant transport problems. This article mathematically elucidates the error in the approximation and presents simulations that compare different versions of the Domenico (1987) solution to an exact analytical solution to demonstrate the potential error inherent in the approximate expressions. Results suggest that the accuracy of the approximate solutions is highly variable and dependent on the selection of input parameters. For solute transport in a medium-grained sand aquifer, the Domenico (1987) solution underpredicts solute concentrations along the centerline of the plume by as much as 80% depending on the case of interest. Increasing the dispersivity, time, or dimensionality of the system leads to increased error. Because more accurate exact analytical solutions exist, we suggest that the Domenico (1987) solution, and its predecessor and successor approximate solutions, need not be employed as the basis for screening tools at contaminated sites.     

Simulating Stable Isotope Ratios in Plumes of Groundwater Pollutants with BIOSCREEN‐AT‐ISO

BIOSCREEN is a well‐known simple tool for evaluating the transport of dissolved contaminants in groundwater, ideal for rapid screening and teaching. This work extends the BIOSCREEN model for the calculation of stable isotope ratios in contaminants. A three‐dimensional exact solution of the reactive transport from a patch source, accounting for fractionation by first‐order decay and/or sorption, is used. The results match those from a previously published isotope model but are much simpler to obtain. Two different isotopes may be computed, and dual isotope plots can be viewed. The dual isotope assessment is a rapidly emerging new approach for identifying process mechanisms in aquifers. Furthermore, deviations of isotope ratios at specific reactive positions with respect to “bulk” ratios in the whole compound can be simulated. This model is named BIOSCREEN‐AT‐ISO and will be downloadable from the journal homepage.     

Domenico solution--is it valid?

The Domenico solution is widely used in several analytical models for simulating ground water contaminant transport scenarios. Unfortunately, many textbook as well as journal article treatments of this approximate solution are full of empirical statements that are developed without mathematical rigor. For this reason, a rigorous analysis of this solution is warranted. In this article, we present a mathematical method to derive the Domenico solution and explore its limits. Our analysis shows that the Domenico solution is a true analytical solution when the value of longitudinal dispersivity is zero. For nonzero longitudinal dispersivity values, the Domenico solution will introduce a finite amount of error. We use an example problem to quantify the nature of this error and suggest some general guidelines for the appropriate use of this solution.     

Evaluating Natural Attentuation with Spreadsheet Analytical Fate and Transport Models

Superposition techniques can extend the capabilities of relatively simple analytical fate and transport models. Complex source geometries, simple aquifer boundaries, and electron acceptor limited biodegradation can be simulated by using superposition techniques in computer spreadsheets. Spreadsheet models are an easily used tool for interpreting sampling results and for estimating attenuation and degradation rates in relatively homogeneous aquifers. Analytical spreadsheet models are based on the Domenico analytical model and can provide results that are in close agreement with the numerical model UIOPLUMH II.     

Contamination Assessment and Site-Management Tool (CAST): A Browser-Based Tool for Site Assessment

Groundwater dependency is increasing globally, while millions of potentially contaminated sites are yet to be characterized for contamination levels. In particular, groundwater contamination due to light nonaqueous phase liquids (LNAPLs) continues to be a global challenge. Mathematical approaches (i.e., analytical, semi-analytical, empirical, numerical) are preferred for an initial site assessment to circumvent the high characterization costs and limited site data availability. However, the site-specific nature of contamination restricts the generalization of any single approach. Hence, the requirement is for an easy-to-use computing interface that provides site-specific data management, the selection and use of multiple-model interfaces for computing, and site characterization, with extension for the latest models as they become available. This work provides one such interface called CAST or Contamination Assessment and Site-management Tool. CAST is an open-source browser-based (online/offline) tool that provides an interface for six different analytical models (e.g., BIOSCREEN-AT), a MODFLOW based numerical model, and two empirical models (including a hybrid numerical-analytical model). Additionally, CAST includes interfaces for site data management, their evaluation, and scenario-based modeling. CAST's development is in a modular format, which simplifies the addition of new computing or data interfaces. Furthermore, the entire code-base of CAST is based on open-source (dominantly Python programming) libraries and standards. This further simplifies the modification or extension of this tool. This paper introduces CAST, its different computing, and data management interfaces and provides examples of the tool's functionality primarily for the initial evaluation of contaminated sites.     

Multidimensional analytical models for isotope ratios in groundwater pollutant plumes of organic contaminants undergoing different biodegradation kinetics

This work presents analytical models which are able to predict contours of concentrations and isotope ratios of organic pollutants in homogeneous aquifers. Four analytical solutions of the advective–dispersive transport equation for reactive transport from the literature differing in assumptions regarding biodegradation kinetics were used. Stable isotope ratios are computed after modelling the individual reactive transport of isotopic species in the aquifer, which respond differently to fractionation by biodegradation or sorption. The main finding of this study is that the isotope ratios in the plumes are very sensitive to the assumptions underlying the biodegradation kinetics in the models. When biodegradation occurs throughout the core of the plume as first-order reaction, the transversal gradients in isotope ratios are smooth. When biodegradation occurs in a bi-molecular reaction with an electron acceptor (modelled by double-Monod kinetics), steep transversal isotope gradients are predicted. When the reaction rates approach instantaneous reaction along the plume fringes, isotope shifts in the core of the plume disappear. A model incorporating plume and fringe degradation produces the most plausible predictions of isotope ratios in this study. It is shown furthermore that isotope fractionation by sorption causes an even different pattern of isotope ratios, with positive shifts restricted to near the forerunning front of an expanding plume. The models developed in this work can serve for the validation of numerical models and may be incorporated in natural attenuation support systems such as e.g. BIOSCREEN.     

Analytical Solutions for Water Flow and Solute Transport in the Unsaturated Zone

Analytical solutions for the water flow and solute transport equations in the unsaturated zone are presented. We use the Broadbridge and White nonlinear model to solve the Richards’ equation for vertical flow under a constant infiltration rate. Then we extend the water flow solution and develop an exact parametric solution for the advection-dispersion equation. The method of characteristics is adopted to determine the location of a solute front in the unsaturated zone. The dispersion component is incorporated into the final solution using a singular perturbation method. The formulation of the analytical solutions is simple, and a complete solution is generated without resorting to computationally demanding numerical schemes. Indeed, the simple analytical solutions can be used as tools to verify the accuracy of numerical models of water flow and solute transport. Comparison with a finite-element numerical solution indicates that a good match for the predicted water content is achieved when the mesh grid is one-fourth the capillary length scale of the porous medium. However, when numerically solving the solute transport equation at this level of discretization, numerical dispersion and spatial oscillations were significant.     

Beitrag zur analytischen Beschreibung von Transportprozessen im Grundwasserleiter bei diffusem Stoffeintrag

A model is described for the relation between water quantity and water quality in the aquifer with the diffuse substance input by the gravitation water being taken into account. The model is based on a partial differential equation of the second order for a flow pipe. For constant transport parameters, a sink of the kinetics of the first order and a time-dependent source term an exact analytical solution is presented and explained by a model. At a diffuse substance input, the dispersion has only a slight influence on the transport of substances contained in water. Potential applications of these solutions to different problems are mentioned.     

Improvements and Corrections to AT123D Code

Although based on exact analytical solutions, semi‐analytical solute transport models can have significant numerical error in applications with high frequency oscillatory source terms and when parameter value combinations cause series solution approximations to converge slowly. Methods for correcting these numerical errors are presented and implemented in the AT123D code, which employs Green's functions to represent point, linear, and rectangular prismatic source zones. In order to increase its computational accuracy, a Romberg numerical integration scheme was added to AT123D with prespecified error criteria, variable time stepping, and partitioning of the integral to handle rapidly changing source terms. More rapidly converging series solution approximations for the Green's functions were also incorporated to improve both accuracy and computational efficiency for finite‐depth aquifers. AT123D also has been modified to eliminate redundant calculations at points where approximate steady‐state conditions have been reached to improve computational efficiency during numerical integration. These modifications help to decrease computer run times that can be excessive for three‐dimensional problems with large numbers of computational points, small time steps, and/or long simulation time periods. Errors in the original AT123D code also were corrected in this modified version, AT123D‐AT, in order to accurately simulate finite‐duration (pulse) source releases.     

Discretizing the fracture-matrix interface to simulate solute transport

This article examines the required spatial discretization perpendicular to the fracture-matrix interface (FMI) for numerical simulation of solute transport in discretely fractured porous media. The discrete-fracture, finite-element model HydroGeoSphere ( Therrien et al. 2005 ) and a discrete-fracture implementation of MT3DMS ( Zheng 1990 ) were used to model solute transport in a single fracture, and the results were compared to the analytical solution of Tang et al. (1981) . To match analytical results on the relatively short timescales simulated in this study, very fine grid spacing perpendicular to the FMI of the scale of the fracture aperture is necessary if advection and/or dispersion in the fracture is high compared to diffusion in the matrix. The requirement of such extremely fine spatial discretization has not been previously reported in the literature. In cases of high matrix diffusion, matching the analytical results is achieved with larger grid spacing at the FMI. Cases where matrix diffusion is lower can employ a larger grid multiplier moving away from the FMI. The very fine spatial discretization identified in this study for cases of low matrix diffusion may limit the applicability of numerical discrete-fracture models in such cases.     

Comments on “Two-dimensional concentration distribution for mixing-controlled bioreactive transport in steady-state” by O.A. Cirpka and A.J. Valocchi

In this comment we present a re-analysis of the analytical solution presented by Cirpka and Valocchi for steady-state concentrations of dissolved bioreactive compounds and bacterial biomass in porous media. We discuss the validity range of the analytical solution. In particular, the criterion used to determine the sustainability of biomass is revisited. This re-analysis shows that the ω$\omega$ criterion used by Cirpka and Valocchi is only a necessary but not a sufficient criterion to determine the bioreactive zones. As a consequence, the analytical solution does not provide the exact distribution of compounds throughout the domain, but can serve as upper or lower boundaries for species concentrations at a given location. These conclusions are supported by the simulation results obtained from an established reactive transport model.     

A three-dimensional analytical tool for modeling reactive transport

In this note, we present a public domain analytical reactive transport modeling tool (ART3D, version 2.0). The tool is developed in FORTRAN and can be used for solving a system of a set of partial differential equations coupled with a first-order reaction network. ART3D uses a novel analytic solution technique proposed by Clement. The new software includes options for performing Monte Carlo simulations and automated parameter estimation.     

Solute Migration from the Aquifer Matrix into a Solution Conduit and the Reverse

A solution conduit has a permeable wall allowing for water exchange and solute transfer between the conduit and its surrounding aquifer matrix. In this paper, we use Laplace Transform to solve a one‐dimensional equation constructed using the Euler approach to describe advective transport of solute in a conduit, a production‐value problem. Both nonuniform cross‐section of the conduit and nonuniform seepage at the conduit wall are considered in the solution. Physical analysis using the Lagrangian approach and a lumping method is performed to verify the solution. Two‐way transfer between conduit water and matrix water is also investigated by using the solution for the production‐value problem as a first‐order approximation. The approximate solution agrees well with the exact solution if dimensionless travel time in the conduit is an order of magnitude smaller than unity. Our analytical solution is based on the assumption that the spatial and/or temporal heterogeneity in the wall solute flux is the dominant factor in the spreading of spring‐breakthrough curves, and conduit dispersion is only a secondary mechanism. Such an approach can lead to the better understanding of water exchange and solute transfer between conduits and aquifer matrix. Highlights:

相似文献   

Saltwater Upconing Due to Cyclic Pumping by Horizontal Wells in Freshwater Lenses

This article deals with the quantification of saltwater upconing below horizontal wells in freshwater lenses using analytical solutions as a computationally fast alternative to numerical simulations. Comparisons between analytical calculations and numerical simulations are presented regarding three aspects: (1) cyclic pumping; (2) dispersion; and (3) finite horizontal wells in a finite domain (a freshwater lens). Various hydrogeological conditions and pumping regimes within a dry half year are considered. The results show that the influence of elastic and phreatic storage (which are not taken into account in the analytical solutions) on the upconing of the interface is minimal. Furthermore, the analytical calculations based on the interface approach compare well with numerical simulations as long as the dimensionless interface upconing is below 1/3, which is in line with previous studies on steady pumping. Superimposing an analytical solution for mixing by dispersion below the well over an analytical solution based on the interface approach is appropriate in case the vertical flow velocity around the interface is nearly constant but should not be used for estimating the salinity of the pumped groundwater. The analytical calculations of interface upconing below a finite horizontal well compare well with the numerical simulations in case the distance between the horizontal well and the initial interface does not vary significantly along the well and in case the natural fluctuation of the freshwater lens is small. In order to maintain a low level of salinity in the well during a dry half year, the dimensionless analytically calculated interface upconing should stay below 0.25.     

Vectorized simulation of groundwater flow and contaminant transport using analytic element method and random walk particle tracking

Groundwater contaminant transport processes are usually simulated by the finite difference (FDM) or finite element methods (FEM). However, they are susceptible to numerical dispersion for advection‐dominated transport. In this study, a numerical dispersion‐free coupled flow and transport model is developed by combining the analytic element method (AEM) with random walk particle tracking (RWPT). As AEM produces continuous velocity distribution over the entire aquifer domain, it is more suitable for RWPT than FDM/finite element methods. Using the AEM solutions, RWPT tracks all the particles in a vectorized manner, thereby improving the computational efficiency. The present model performs a convolution integral of the response of an impulse contaminant injection to generate concentration distributions due to a permanent contaminant source. The RWPT model is validated with an available analytical solution and compared to an FDM solution, the RWPT model more accurately replicates the analytical solution. Further, the coupled AEM‐RWPT model has been applied to simulate the flow and transport in hypothetical and field aquifer problems. The results are compared with the FDM solutions and found to be satisfactory. The results demonstrate the efficacy of the proposed method.     

Estimation of Degradation Rates by Satisfying Mass Balance at the Inlet

Using a steady-state mass conservative solute transport analytical solution that is based on the third-type (or flux-type or Cauchy) source condition, a method is developed to estimate the degradation parameters of solutes in groundwater. Then, the inadequacy of the methods based on the first-type source-based analytical solute transport solution is presented both theoretically and through an example. It is shown that the third-type source analytical solution exactly satisfies the mass balance constraint at the inlet location. It is also shown that the first-type source (or constant source concentration or Dirichlet) solution fails to satisfy the mass balance constraint at the inlet location and the degree of the failure depends on the value of the degradation as well as the flow and solute transport parameters. The error in the first-type source solution is determined with dimensionless parameters by comparing its results with the third-type source solution. Methods for estimating the degradation parameter values that are based on the first-type steady-state solute transport solution may significantly overestimate the degradation parameter values depending on the values of flow and solute transport parameters. It is recommended that the third-type source solution be used in estimating degradation parameters using measured concentrations instead of the first-type source solution.     

The beneficial role of rubble mound coastal structures on seawater oxygenation

The beneficial role of rubble mound coastal structures on oxygenation under the effect of waves is discussed, based on analytical considerations and experimental data from laboratory experiments with permeable and impermeable structures. Significant oxygenation of the wave-protected area was observed as a result of horizontal transport through the permeable structure. A two-cell model describing the transport of dissolved oxygen (DO) near a rubble mound breakwater structure was developed and used for the determination of the oxygen transfer coefficients from the experimental data. Oxygen transfer through the air-water interface is considered a source term in the transport equation and the oxygen flux through the structure is taken into account. The mass transport equations for both sides of the structure are solved analytically in terms of time evolution of DO concentration. The behaviour of the solution is illustrated for three different characteristic cases of initial conditions. The oxygen transfer through the air-water interface in the wave-influenced area increases the DO content in the area; the resulting oxygen flux through the structure is discussed. The analytical results depend on the initial conditions, the oxygen transfer coefficient and the exchange flow rate through the structure. Experiments with impermeable structures show that air water oxygen transfer in the harbour area is negligible in the absence of waves. In addition the ratio of the horizontal DO flux to the vertical flux into the seaward side tends towards a constant value, independent of the initial conditions.     

Using PHREEQC to simulate solute transport in fractured bedrock

The geochemical computer model PHREEQC can simulate solute transport in fractured bedrock aquifers that can be conceptualized as dual-porosity flow systems subject to one-dimensional advective-dispersive transport in the bedrock fractures and diffusive transport in the bedrock matrix. This article demonstrates how the physical characteristics of such flow systems can be parameterized for use in PHREEQC, it provides a method for minimizing numerical dispersion in PHREEQC simulations, and it compares PHREEQC simulations with results of an analytical solution. The simulations assumed a dual-porosity conceptual model involving advective-reactive-dispersive transport in the mobile zone (bedrock fracture) and diffusive-reactive transport in the immobile zone (bedrock matrix). The results from the PHREEQC dual-porosity transport model that uses a finite-difference approach showed excellent agreement compared with an analytical solution.     

Analytical and numerical solutions for wave propagation in water-saturated porous layered half-space

A fluid-saturated one-layer continuum underlain by a rigid half-space is considered. An exact solution is developed in frequency domain for analyzing disturbance induced by a strip footing located at the surface with vertical harmonic excitation. Since the analytical solution can be used only for very simple conditions, a finite element model has been developed also and compared with the exact solution. It is shown that finite element results are in close agreement with the results which have been obtained by a transformation technique. The proposed finite element scheme can take into account the complex geometry and inhomogeneity for practical problems. Besides this, the analytical results exhibit the overall characteristic of wave propagation in porous media and will provide a representative test problem which can be used for a quantitative evaluation of the accuracy of various numerical solution methods.     

AN ANALYTICAL SOLUTION FOR CALCULATING THE INITIATION OF SEDIMENT MOTION

This paper presents an analytical solution for calculating the initiation of sediment motion and the risk of river bed movement. It thus deals with a fundamental problem in sediment transport, for which no complete analytical solution has yet been found. The analytical solution presented here is based on forces acting on a single grain in state of initiation of sediment motion. The previous procedures for calculating the initiation of sediment motion are complemented by an innovative combination of optical surface measurement technology for determining geometrical parameters and their statistical derivation as well as a novel approach for determining the turbulence effects of velocity fluctuations. This two aspects and the comparison of the solution functions presented here with the well known data and functions of different authors mainly differ the presented solution model for calculating the initiation of sediment motion from previous approaches. The defined values of required geometrical parameters are based on hydraulically laboratory tests with spheres. With this limitations the derivated solution functions permit the calculation of the effective critical transport parameters of a single grain, the calculation of averaged critical parameters for describing the state of initiation of sediment motion on the river bed, the calculation of the probability density of the effective critical velocity as well as the calculation of the risk of river bed movement. The main advantage of the presented model is the closed analytical solution from the equilibrium of forces on a single grain to the solution functions describing the initiation of sediment motion.